Abstract
Mean motion resonances (MMR) are a frequent phenomenon among extrasolar planetary systems. Current observations indicate that many systems have planets that are close to or inside the 2:1 MMR, when the orbital period of one of the planets is twice the other. Analytical models to describe this particular MMR can only be reduced to integrable approximations in a few specific cases. While there are successful approaches to the study of this MMR in the case of very elliptic and/or very inclined orbits using semi-analytical or semi-numerical methods, these may not be enough to completely understand the resonant dynamics. In this work, we propose to apply a well-established numerical method to assess the global portrait of the resonant dynamics, which consists in constructing dynamical maps. Combining these maps with the results from a semi-analytical method, helps to better understand the underlying dynamics of the 2:1 MMR, and to identify the behaviors that can be expected in different regions of the phase space and for different values of the model parameters. We verify that the family of stable resonant equilibria bifurcate from symmetric to asymmetric librations, depending on the mass ratio and eccentricities of the resonant planets pair. This introduces new structures in the phase space, that turns the classical V-shape of the MMR, in the semi-major axis versus eccentricity space, into a sand clock shape. We construct dynamical maps for three extrasolar planetary systems, TOI-216, HD27894, and K2-24, and discuss their phase space structure and their stability in the light of the orbital fits available in the literature.
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Notes
The notation \(\Delta \sigma _i\) refers to the libration amplitude of a specific orbit, while \(\delta \sigma _i\) denotes the resonance width. Therefore, \(\max \Delta \sigma _i\approx \delta \sigma _i\).
This law can be described approximately by:
$$\begin{aligned} e_1=\frac{C_1m_2}{M_{\star }\displaystyle \left( \frac{P_2}{P_1}-\beta \right) } \end{aligned}$$(3)where \(C_1\) and \(\beta \) are constants and \(M_{\star }\) is the stellar mass (Charalambous et al. 2017). For this particular case, \(C_1=1.2\) and \(\beta =2.01\).
The asymmetric librations for \(m_2/m_1<1\) occur already in the framework of the restricted circular three body problem (Beauge 1994).
Recall that in the restricted circular three body problem, the resonance width scales as \(\sqrt{m_\textrm{pert}}\).
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Acknowledgements
The simulations presented in this work have been run at the SDumont and LoboC clusters of the Brazilian System of High-Performance Computing (SINAPAD), and at the Mulatona Cluster from the CCAD-UNC, which is part of SNCAD-MinCyT, Argentina. FR wishes to acknowledge support from the Brazilian National Council of Research (CNPq). This work has been partially financed by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
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Appendix A: Additional maps for TOI-216 and HD27894
Appendix A: Additional maps for TOI-216 and HD27894
Maps of the MEGNO chaos indicator for the TOI-216 system in the plane \(a_1, e_1 \cos (\Delta \varpi \)), computed over a simulation time span of 1000 years (left) and 10,000 years (right). These shall be compared with the fourth panel in Fig. 3, that was computed over 100 years
Dynamical maps for the TOI-216 system in the plane \(a_1, e_1 \cos (\Delta \varpi \)), showing the variations of the more massive planet, \(\max \Delta \sigma _2\) and \(\max \Delta e_2\). Panels shall be compared to Fig. 3. It is noteworthy the libration of \(\sigma _2\) for values of \(e_1\gtrsim 0.15\), which, combined with the libration of \(\sigma _1\), is consistent with the presence of ACRs in that region (e.g., Michtchenko et al. 2008a)
Dynamical maps for the HD27894 system in the plane \(a_2, e_2 \cos (\Delta \varpi \)), showing the variations of the more massive planet, \(\max \Delta \sigma _1\) and \(\max \Delta e_1\). Panels shall be compared to Fig. 8. The libration of \(\sigma _1\) observed here for the whole range of \(e_2 \cos \Delta \varpi <0\), combined with the libration of \(\sigma _2\) about \(e_2 \cos \Delta \varpi \sim -0.02\), implies in the occurrence of ACRs consistent which the theoretical predictions for the mass ratio considered (e.g., Michtchenko et al. 2008b)
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Giuppone, C., Rodríguez, A., Alencastro, V. et al. Mapping the structure of the planetary 2:1 mean motion resonance: the TOI-216, K2-24, and HD27894 systems. Celest Mech Dyn Astron 135, 3 (2023). https://doi.org/10.1007/s10569-022-10112-5
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DOI: https://doi.org/10.1007/s10569-022-10112-5